ruth-value realism is a position that has in common with mathematical Platonism a degree of realism, insofar as according to this position, mathematical statements and formulas are objectively true. What distinguishes this position from mathematical Platonism, however, is that “it is not committed to the distinctively platonist idea that these truth-values flow from an ontology of mathematical objects”(Linnebo 2011). It ought to be noted, however, that there are nominalist-leaning ontologies ofmathematics that are nonetheless compatible with forms of truth-value realism.
Working realism is a kind of mathematical pragmatism according to which we ought to practice mathematics as though mathematical Platonism were true, even though it might not be. Those who advocate working realism bracket ontological debates about the metaphysics of mathematics and emphasize its methodological component. Plenty of working realists are mathematical platonists, of course, and many of them argue that the pragmatic efficacy of working realism provides strong support for mathematical Platonism.
Working realists oftentimes argue, in the vein of Platonism, that we would not need to restrict our methodology, formula, axioms and so on, if they did not in some sense really correspond to reality. Gödel makes the point that “impredicative definitions are legitimate whenever the objects being defined exist independently of our definitions”(Linnebo 2011). For example, if I say that a hamburger is such and such pounds, such an impredicative definition is legitimate provided it has as its referent an actual hamburger. Furthermore, if mathematics were objectively independent, it rather easily explains how each and every math problem has a unique answer (this is related to what we noted earlier about the necessity of restriction in formulas and axioms)(Linnebo 2011). This seems to support the Hilbertian optimism according to which every mathematical problem has a solution(Linnebo 2011).
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